Genesis of Eden

Abstract : This paper presents a review of fractal and chaotic
dynamics in nervous systems and the brain, exploring mathematical
chaos and its relation to processes, from the neurosystems level
down to the molecular level of the ion channel. It includes a
discussion of parallel distributed processing models and their
relation to chaos and overviews reasons why chaotic and fractal
dynamics may be of functional utility in central nervous cognitive
processes. Recent models of chaotic pattern discrimination and
the chaotic electroencephalogram are considered. A novel hypothesis
is proposed concerning chaotic dynamics and the interface with
the quantum domain.

This review surveys fractal and chaotic processes in brain
dynamics and provides workers in experimental fields with a compact
source of material in mathematical chaotic dynamics as a reference.
An attempt has been made to make the mathematical aspects of the
paper remain approachable to a variety of readers. Full background
references are given to enable the reader to gain further in-depth
treatment, and to explore more fully the variety of specialist
topics leading out from the discussion.

Section 1 provides a compact mathematical introduction to fractal
and chaotic dynamics. Most of the systems discussed here have
specific application to experimental results in later sections.
Sections 2 and 3 complement this with source material on mathematical
modelling of neural nets and on biological neurons. In section
4 chaos at the cellular level is discussed, including models of
the excitable membrane and ion channel. In section 5 chaotic neurosystems
models and experimental results are considered including the Freeman-Skarda
model and studies of the EEG. Section 6 touches on issues connecting
quantum chaos, causality and the mind.

The aim of the review is to make it possible for a reader to
gain a comprehensive overview of chaos as it applies to neural
processes, to compare chaotic models with their alternatives and
to assess the scope of chaotic and fractal processes in the conceptualization
of the physical basis of brain function.

1: Concepts and Techniques in Chaos

(a) Chaotic systems This
introduction provides a general description of chaotic dynamics
and outlines techniques useful in analysis of experimental results.

A dynamical system is defined to be a set of first-order differential
equations :

[1.1]

The system is called autonomous if , i.e. the functions
are not changing with time. Systems of higher-order differential
equations can be reduced to first-order systems by a suitable
change of variables. For example the equation of harmonic motion
can
be simply reduced to the system . This type of
formulation is general to Hamilton representations of conservative
energetic systems, (see [2.2]).

A dynamical system is generally defined on a configuration
space consisting of a topological manifold. A manifold is locally
like Euclidean space, but may have varied global
structures, as exemplified by the cylinder, torus, fig 5(a), Klein
bottle and other higher-dimensional spaces.

Integration of the equations [1.1] to [1.2]

yields integral curves or trajectories forming a flow on the
manifold. The sets of these flow curves are called orbits. The
flow thus integrates the field of velocity vectors determined
by [1.1].

Linear differential equations admit analytic solutions and
have well-defined asymptotic behavior as , converging to
fixed points, or periodic oscillations, forming closed orbits.
By contrast, even the simplest deviations from linearity, including
quadratic, bilinear and piecewise-linear functions can, under
suitable conditions, result in more complex chaotic behavior,
in which the orbits of the system are attracted to a complex higher-dimensional
subset called a strange attractor, or are ergodic. Ergodic flows
behave like thermodynamic systems in that they can be modelled
over statistical ensembles because the orbit fills a (possibly
dense) set of invariant measure.

An attractor is a subset of the manifold to which an open subset
of points, the basin of the attractor, tends in limit with increasing
time. For example in fig 1(b) for v = +1 the system has an attractor
consisting of the closed circular orbit, with basins outlined
by the arrowed flows. Existence of an attractor requires local
volume to contract with increasing time and is consistent with
a dissipative system in position-momentum representation. While
conservative systems thus do not have attractors, they may still
display classical chaos, see section 1(c)(iii).

Fig 1 : (a) The Lorentz system displays sensitive
dependence in which neighboring trajectories separate exponentially
with time. Neighboring trajectories emanating from a are visibly
separate by b and diverge into distinct spirals by c1 and c2,
so that their subsequent dynamics is unrelated. (b) The Hopf bifurcation
forms a periodic closed orbit attractor. For v < 0
there is only a sink (attractor). As v crosses 0 an attracting
periodic orbit and source (repellor) are created. The basins are
indicated by the arrows. (c) Geometric profile of the improved
Euler method [1.4].

The hall-mark of a chaotic flow is sensitive dependence on
initial conditions (Schuster 1986). Points which are arbitrarily
close initially become exponentially further apart with increasing
time, leading to the amplification of very small perturbations
into global uncertainties. Sensitivity results in both an entropy
increase associated with the loss of positional information with
time, and in structural instability in which an arbitrarily small
perturbation of the flow causes structural changes to the topology
of the orbits ( although they may have similar qualitative behavior
). This prevents accurate long-term numerical approximation of
the system with increasing time. In fig 1(a) sensitive dependence
is illustrated. Neighboring trajectories emanating from a are
visibly separate by b and diverge into distinct spirals by c1
and c2, so that their subsequent dynamics is unrelated. Note that
each orbit has a unique winding sequence e.g. 1, 6, 3, 4, ...
representing the number of times it negotiates each spiral arm
of the attractor. An arbitrarily small perturbation will disrupt
the winding sequence and hence change the topology of the orbits.
The entropy results in a loss of memory of the initial conditions
in any numerical approximation over time. The initial conditions
thus cannot be retrieved by reverse iteration of the flow.

Time-dependent systems are capable
of abrupt changes in their topological form called bifurcations
as the underlying parameters cross critical values. Bifurcations
result in abrupt catastrophic change in the topology of the flow
under continuous variation of the time-dependent parameters. In
fig 1(b) the Hopf bifurcation results in the formation of a closed
orbit attractor (oscillation) from a point attractor (sink) at
the origin as v crosses 0, see section 1(c)(ii). In fig 4(a) repeated
pitchfork bifurcations result in subdivision of the logistic attractor,
and the tangent bifurcation results in intermittent chaos, section
1(c)(i).

Because non-linear differential equations cannot in general
be integrated directly, it is often necessary to resort to techniques
of numerical integration in which a discrete transfer function
is constructed which approximates a stroboscopic representation
of the flow at discrete time intervals

[1.3]

by using numerical methods such as the improved Euler method
[1.4] or Runge-Kutta (Butcher 1987) :

[1.4]

In a time-varying system, chaos may become established by three
principal routes involving a (possibly infinite) sequence of bifurcations
of the attractor, intermittent disruption of a periodicity, or
the topological breakup of a surface, such as a torus, representing
several linked oscillations. We will examine each of these three
routes, because a knowledge of all of them is essential in characterizing
chaotic dynamics in the brain and excitable cells.

(b) Indicators of Chaos

A series of techniques have been developed for analysing chaotic
systems which both lead to a conceptual understanding of their
phenomenology and also provide methods for handling experimental
investigations. These are outlined in the following sections.

(i) Liapunov Exponent and
Entropy. Two of the most important attributes of chaotic
systems are sensitive dependence on initial conditions and the
loss of spatial information with time, resulting in an entropy.

In a flow with sensitive dependence the distance between adjacent
points becomes exponentially further apart with increasing time.
Repeated iteration of the corresponding chaotic map similarly
causes the separation of two adjacent points to become exponentially
increased. This provides a means of calculating the exponent of
growth, called the Liapunov exponent.

Consider the repeated action of the discrete map increasing
separation by a factor :

Hence we can write .

If the separation varies along the path, we can take limits
as
we have

[1.5]

[1.6]

This formula makes it easy to calculate the Liapunov exponent
for any iteration. In a chaotic system in one or more variables,
sensitive dependence requires at least one of the Liapunov exponents
to be greater than 1, thus resulting in exponential separation
of trajectories.

Note that in the case of a continuous flow, the role of the
constant a is slightly different.

In the flow , [1.7a] whereas with the map, [1.7b]

The formula [1.6] also naturally represents the loss of information,
or entropy :

The Shannon informational entropy is [1.8]

where is the probability of being in state
i and .

Consider a single iteration in which [0,1] maps to [0,1] under
separation a.. At the initial stage, we have n states each
with probability , so : [1.9]

After one iteration, the resolution is reduced by factor ,
following the same reasoning as in [1.6] for one step, giving
states
each with probability ,

so we have : [1.10]

Thus there is thus a difference .

Averaging this over many iterations, we have [1.11]

which is obviously the same as [1.6] except for a factor of
log 2.

For a 1-dimensional map we thus define the Kolmogorov entropy
to be K = l.

When we have a higher-dimensional mapping or flow, there is
an exponent li for each dimension i in the configuration space.
If the system has an attracting set, volume contraction will cause
the sum of the exponents to be negative, thus allowing only some
to be positive. Only the positive (expanding) Liapunov exponents
contribute to the spreading and so K is generally identified with
the sum of these positive exponents: [1.12]

A system with more than one positive exponent is referred to
as hyperchaotic (Rössler 1979,1988).

From another point of view, the entropy may be associated with
new information entering the system and over time replacing that
associated with the initial conditions.

(ii) Power Spectrum.
To distinguish between chaotic and multiply periodic systems one
can examine the Fourier transform [1.13]

which transforms the function x(t) into a spectrum x(w) of
frequency components, which for periodic motion consists of discrete
frequencies, but for chaotic motion has a broad band spread.

The power spectrum squares the Fourier amplitudes to give positive
real values

[1.14]

In the case of a discrete iteration of finite length, such
as a 2n cycle, we can use a discrete transform to resolve the
iteration into its Fourier components [1.15]

This results for example in the 1024 step and 256 step Fast
Fourier Transforms [FFTs] shown in fig 2(a). In (i) a 1024 point
iteration of one component of the Lorentz iteration has been used
to generate the discrete power spectrum using a Fast Fourier Transform
[FFT] (Elliot et. al.. 1982). Although the flow has a strong periodicity
its band-spread indicates chaos. (ii) The FFT of the self-similar
Morse-Thule sequence (Schroeder 1986) 0110100101101001. . . This
can be generated by recursive reflection of 01 in its complement
[viz 01, 0110, 01101001, etc.], or by taking the binary digit
sums of each positive integer mod 2, [viz 0 Æ 0, 1 Æ
1, 10 Æ 1, 11 Æ 0 etc.]. Although this is non-repeating,
any discrete Fourier transform has a symmetric set of distinct
components, as a result of symmetries in the self-similar structure.

Fig 2 : (a) Power spectra (i) for the Lorentz
system and (ii) for the Morse-Thule sequence. Note the broad bandspread
in (i) characteristic of chaos despite the existence of a peak
frequency. In (ii) although the sequence is self-similar and non-periodic,
the power spectrum consists of symmetrical frequencies. (b) Koch
flake formed by repeated tesselation of a triangle and (c) Cantor
set formed by repeated removal of the central third of each interval
are examples of fractals. (d) Julia set of complex logistic map
[1.24] for r = (3.4 + .02 i). This set is connected so that points
near zero iterate to a finite period attractor inside the set.
Points outside all iterate to infinity. Julia sets can also be
disconnected, in which case all other points iterate to infinity.

(iii) Hausdorff Dimension and Fractals. In a
one-dimensional set such as the interval [0,1], we need twice
as many subsets of 1/2 the length to cover the interval, 4 times
as many of a 1/4 the length and so on. In a two dimensional set,
such as the unit square, we need times as many
1/2 the length and so on. We can thus define the Hausdorff dimension
d as the exponent such that a covering by d-spheres of diameter
satisfies , as [1.16]

A set is called a fractal if its Hausdorff dimension exceeds
its integral topological dimension, i.e. if the Hausdorff dimension
is not an integer.

Fractals often possess self-similarity on a change of scale
between parts of the set and the whole. A fractal which is constructed
by recursive development in stages, enables exact calculation
of the Hausdorff dimension from [1.16] using two successive stages
of length and : [1.17]

For example in the Koch flake, fig 2(b) each side is repeatedly
replaced by 4 sides each 1/3 the length. This gives rise to a
Hausdorff dimension of in the limit, hence a fractal.
The Cantor set, fig 2(c) is formed from [0,1] by removing the
open middle third [e.g. ] from each remaining subinterval,
leaving 2 sides of the length 1/3.

It is thus a fractal with dimension . Note that the
Cantor set is identifiable with all base 3 numbers in [0,1] having
only 0 & 2 as digits, e.g. 0.0220020. . . and thus maps 1­1
onto the whole interval [0,1] by mapping e.g. 0.0220020
0.0110010. This occurs despite the removal of a set of measure,
equal to that of the whole interval [0,1]!

The strange attractors of chaotic dynamics are generally fractals.
A finite non-integer dimension indicates that a dynamic is chaotic,
rather than stochastic. Iterations such as the logistic map [1.23]
result in fractals called Julia sets fig 2(d) , subsets which
tend neither to a finite attractor nor to infinity, but are mapped
within themselves. They thus form the fractal boundaries of the
basins of attraction. The variety of Julia sets of the quadratic
mapping has been the subject of keen interest (Peitgen
& Richter 1986), as well as their relation to the Mandelbrot
set fig 4(a). The Julia set of each value r is a unique self-similar
fractal each with its own distinctive form. Given a connected
Julia set, such as the one illustrated in fig 2(d), for the period
2 region of the logistic map, points on the interior basins all
iterate to a finite attracting set, while those outside iterate
to infinity. The bounding Julia set consists of those chaotic
points which do neither. Such fractal basin boundaries appear
to be general in dynamical systems. Time-dependent systems may
also generate fractal spatial bifurcations to form dissipative
structures enabling the generation of fractal order out of chaos.

(iv) Correlation Integral:
Generalized Dimensions We can generalize the fractal dimension
as follows (Grassberger & Procaccia 1983, Roschke & Basar
1989). Consider a covering, as above, with spheres of radius e,
Pi the probability that a point falls into sphere i, and
the number of non-empty spheres. Then we define the Réngi
information of order q as :

where is the probability of points having distance
,
since each is the probability of being
in the same -sphere. can be calculated explicitly
using the Heaviside step function.

[1.21]

Since , we have [1.22]

making it possible to do a plot of log against
log,
to test for linearity, fig 3(c).

The correlation dimension is thus a more accessible measure
of the dimension of a chaotic attractor than the fractal dimension,
which is more difficult to calculate from a trajectory partly
because the points do not become evenly spread on the attractor.
Dimensions vary from 2.06 for the Lorentz attractor, through 4
for e.e.g. recordings of epileptic states through to values around
9 for a stochastic process with a small degree of correlation
between the sample variables.

Because a long time series of vectors x1,
... , xn in the dynamic will have most
of its variables uncorrelated because of exponential divergence
of the trajectories, correlations between the variables will be
a consequence of their lying on the attractor. Modifications of
the Grassberger-Procaccia algorithm have been proposed (Theiler
1987, Albano et. al. 1988, Rapp et. al. 1989) which improve both
speed and accuracy.

As a result of Taken's (1981) proof, a 1-D time series can
be used to form an embedding space for the attractor by taking
k-vectors xit, x(i+1)t,
... , x(i+k-1)t , by taking a suitably
large value for the dimension k. Increasing the time delay t results
in a saturation level tsat for a given k as the sampling time
becomes long enough to ensure non-correlation, fig 3(a). As k
is increased, tsat increases to a plateau, thus determining suitable
k, t and hence d2 (Roschke & Basar 1989), fig 3(b). Various
problems still remain. A suitable choice of the window length
(k-1)t and of the overall time epoch of the sample must be made.
A good measurement of d2 is made only if the slope of the log-log
plot remains say within a 10% variation over at least an adjustment
of a factor of 2 in e (Rapp et. al. 1989). A plot of the slope
against log e is useful here. The upper and lower bounds defining
the region should be indicated. One reasonable indicator of window
length is to use the autocorrelation function [1.23]

to define the correlation time when the autocorrelation function
has fallen from 1 at t=0 to 1/ (Albano et.al.
1988).

Fig 3 : The experimental determination of correlation
dimensions requires testing of parameters for saturation. (a)
tsat versus k, as
different embedding dimensions are used and the delay k is increased,
saturation occurs in the estimated dimension for different delays.
Adequate delay must be provided in each embedding dimension to
get a correct figure. (b) plateau in tsat at k = 7, illustrates
that adequate embedding dimension is also essential. (c) d2 from
slopes of log C(e)-log e plots has limit at k = 7. The dimensions
are actually calculated from the limiting slope of log C(e) against
log e. Checks should be made both that the parameters used do
give linearity in the parameter range and that they do converge
to a good limit value.

A good plateau in the slope depends on a suitable window length
a few times larger then the correlation time. Too short a window
fails to provide a good plateau, on the other hand, making the
window too large can result in the values no longer strictly adhering
to a single trajectory and violating Taken's embedding theorem (updated by jody).
Similarly care had to be taken with the time-epoch, which should
be as long as possible but not long enough to result in non-stationarity
in the phenomenon being measured. Researchers devising experimental
tests for chaos are advised to consult Rapp (1989) before choosing
their design and protocol. See also section 1(d).

It is also possible to measure the correlation dimension by
taking a series of measurements from distinct spatial points in
the dynamic, however this may result in a lowering of the measured
value because the presentation of the attractor from the spatial
sample is not fully unfolded (Babloyantz 1989).

(c) Iterations as Examples of
Chaos

(i) The Logistic Map.
Chaotic dynamics occur in some of the simplest iterative functions,
including piecewise-linear and quadratic functions. To develop
several aspects of chaotic behavior, we will examine a typical
quadratic iteration, the logistic map :

[1.24]

representing exponential population growth subject to a constrained
area (Schuster 1986, Devaney & Keen 1989). The term r xn determines of exponential growth while the
additional term (1 - xn) places a finite
area constraint limiting the population.

It is easy to picture such an iteration in various ways. One
is to successively evaluate the functions y = r x (1 - x) and
x = y as shown in fig 4(bi). We pick an initial value x and evaluate
y by moving vertically to the parabola. Next we let x = y by moving
horizontally to the sloping line. The two steps combined result
in one iteration i.e. xn+1 = y = r xn (1 - xn).

As the parameter r varies, the behavior of the iteration goes
through a sequence of different stages. In (bi) the iterations
are illustrated for r = 1 and 2 starting from two arbitrary points
in [0,1]. Each iterates toward a fixed point, one at zero and
the other positive. For the remaining figures the iteration is
left to run for a few hundred steps, before plotting, so that
only the limiting attractor is highlighted. Near the value 3.4
the iteration is attracted to a set of two values i.e. period
2, as depicted in (bii), in which the arrows still indicate the
y = r x (1 - x) and x = y steps. At 3.56, (biii), the period 2
orbit has bifurcated twice to form a period 8 orbit. The effect
of such period doubling is is clearly seen in the braided form
of the attractor path. At 3.66, (biv), chaos has set in and the
orbits now spread irregularly across the interval without returning
exactly. At 3.8282 (bv) we are very close to the period 3 window.
The period 3 iteration keeps slipping however, and intermittently
enters chaos before returning to the attractor. At 3.8289 (bvi)
period 3 has become stable. At 4.5 (c) the attractor has broken
up and now most points escape to ­ . A residual Cantor set
of points (the Julia set) is mapped amongst itself, forming a
Smale horseshoe (see section 1(c)(iii)).

Alternatively, we can plot all the x values that occur for
a given r, once the system has been allowed to approach the attractor,
as shown in fig 4(a). This gives rise to the attractor form diagram
in which an initial point attractor repeatedly bifurcates into
2, 4, 8, ... values limiting in chaos at r, punctuated by further
windows e.g. of period 3, and finally breakup of the attractor
at r = 4.

Fig 4 : The logistic map : The text contains a
complete description of all the phenomena in the diagram. (a)
The forms of the attractor, Liapunov exponent and Mandelbrot set,
for 2.8 < r < 4 showing the development
of multiple period doublings, chaotic regions and periodic windows.
The attractor initially is a single curve (point attractor) but
then repeatedly subdivides (pitchfork bifurcations) finally entering
chaos (stippled band). Subsequently there are windows of period
3, 5 etc. with abrupt transitions from and to chaos caused by
intermittency and crises. The Liapunov exponent< 0 until chaos sets in. During chaos it remains
positive. The Mandelbrot set illustrates the fractal nature of
the periodic and chaotic regimes when x & r are extended to
the complex number plane. Complex number representation aids visualizing
such fractal structures. (b) A series of 2-D iterations of Gr(xn)
including periods 1, 2 and 8 chaos, intermittency, and period
3. In (i) the two-step iteration process is illustrated alternately
evaluating y = r x (1 - x) (vertical) and x = y (horizontal).
As r crosses the value 1 a saddle-node bifurcation occurs resulting
in the attractor moving from zero and leaving a repellor there
(r = 2). In (ii) & (iii) period 2 and 8 attractors have formed.
In (iv) the iteration has become chaotic. In (v) the chaos is
intermittently entering a period 3 regime, which has become stable
in (vi). (c) The Cantor set of the horseshoe for r = 4.5. The
attractor has now broken up resulting in most points iterating
to minus infinity, leaving only a Julia set of exceptional points.
(d) The pitchfork bifurcation illustrated. The double iterate
Gr^2(xn) twists to cross y = x an extra time, resulting in doubling
of the attractor into a period 2 set. (e) The tangent bifurcation
illustrated using the triple iterate Gr^3(xn). The lifting of
the central tangent above y = x removes the stability of period
3 causing slippage and intermittent chaos.

The sector of the Mandelbrot set of the logistic map fig 4(a)
illustrates the fractal nature of the envelope of iterates when
both x and r have complex number values. These enable us to visualize
the fractal structures more easily because they form a plane.
Each r value in the Mandelbrot set gives rise to a connected Julia
set fig 2(d) and will hence iterate the central value x = 1/2
to a finite attractor separated from infinity by the connected
Julia set (e.g. inside the Julia set of fig 2(d)). The complement
of the Mandelbrot set will iterate 1/2 to infinity. In fig 4(a)
the Mandelbrot set becomes vertically extensive only for r values
whose real part has Liapunov exponent < 0. For > 0
it is confined to the real line, extending a thread to the value
4 with tiny islands showing for r values in the odd period windows.
The Mandelbrot set, and particularly its complement near their
boundary, is famous for the beauty of its color contour computer
iterations. It has been described as the most complex object in
mathematics.

We will examine the variety of dynamical phenomena which occur
in the Logistic map by looking qualitatively at six situations,
each of which highlights a distinct feature of importance :

(1) Point attractors : When r = 0 the attractor is initially
zero. As the parameter r is increased from 0, the quadratic rises
and at 1 crosses the line y = x resulting in a saddle-node bifurcation
in which a single attractor becomes a pair : an attractor and
a repellor. In higher dimensional situations we would have a saddle,
fig 6(c) and an attractor or repellor (node). The point attractor
moves up to positive x, leaving a repellor at 0. Outside [0,1]
the iteration tends to minus infinity. This situation is illustrated
in fig 4(bi) where for the transitional value r = 1 the iteration
is still attracted down and to the left to zero, while for r =
2, zero is a repellor and the intersection of the parabola with
the line y = x is an attractor.

(2) Period doubling : At value r1 ~ 3 there is a bifurcation
of the fixed attractor into a period 2 attracting set, as illustrated
in (bii). Successive period doublings at r2etc.,
(bii, iii) cause the attractor to have a sequence of periods 2,
4, 8, ... , 2n. These arise from pitchfork bifurcations as illustrated
obviously in the forkings of attractor form in (a). Here the graph
of the two-step iterate Gr^2(x) = Gr(Gr(x)) twists across
y = x to cause a doubling of the period. In this range the Liapunov
exponent < 0, except at r1 , r2 , etc. where =
0.

where dn are the widths of the period 2^n attractors where
they straddle the symmetrical value 1/2.

The values [1.25]

are determined by the Feigenbaum numbers d = 4.669, and =
2.502. These are universal to all functions with a quadratic maximum
and thus appear in a variety of systems from biology to astronomy
(Stewart 1989).

(3) Chaos : At the limit value r the iteration becomes
chaotic, (biv) and > 0. The trajectories now
spread over the interval [0,1]. They do not recur as there are
no finite periodicities, but approach each possible value arbitrarily
closely given sufficient time. The iteration now has sensitive
dependence, e-close initial points becoming exponentially separated.
Although the orbits appear equally spread across the entire possible
range of values, the details of each are structurally unique.
Complexity grammars (Auerbach & Procaccia (1990) further analysis.

(4) Odd Period Windows : Intermittency and Crises There are
a series of windows in the chaotic region where chaotic behavior
is abruptly interrupted by new periodic regimes of periods 3,
5, etc., (bvi). These windows contain for example 3.2n bifurcation
sequences similar to that of (2). As a result of Li & Yorke
(1975), the existence of a period 3 attractor guarantees the existence
of periods of all orders and uncountably many aperiodic orbits
(chaos). By Sarkovski, the periods follow the causal sequence
:

3 5 7 ... 2n.3 2n.5
... 2^4 2^3 2^2 2 1,
n = 1, 2, 3,...

At the left-hand end of the period 3 window, a new type of
bifurcation, the tangent bifurcation occurs, in which the 3-cycle
becomes intermittently disrupted by chaotic bursts, (bv). Intermittent
disruption of a periodic dynamic constitutes a second route to
chaos distinct from period doubling in which only a single bifurcation
is required for chaos. These constitute two of the three classical
route to chaos. In fig 4(e) the source of the tangent bifurcation
is illustrated. The tangent to Gr^3(x) crosses y = x as r is decreased
allowing the escape of the period 3 iteration. Immediately upon
bifurcation, the tangent (upper fig) is adjacent to y = x causing
a slow slippage of the 3-cycle with irregular breakout into short
episodes of chaos. Hence the term intermittency.

At the right-hand end of the period 3 window is another type
of abrupt transition to chaos called a crisis that is caused by
a collision between a point repellor and the fanning chaotic sub-bands
of period 3 forming small triangles in fig 4(a). This causes the
chaos to be repelled so that it spreads suddenly across all values
again. The 3 repellors originate from the birth of period 3 in
the tangent bifurcation at the other end of the window.

(5) Julia sets and Horseshoes : For each value of r there is
a residual fractal Julia set of exceptional points which do not
converge to the attractors, but are mapped instead among themselves.
The Julia set for an r value of the complex logistic map in the
period 2 region is illustrated in fig 2(d). Complex values assist
the visualization of Julia sets because complex numbers form a
planar image which we can see.

For r > 4 the finite attractor ceases to exist,
since the graph now goes outside the unit square, allowing points
to iterate to minus infinity, however a Cantor set of points remains,
fig 4(c) which are mapped among themselves indefinitely, once
all the points which escape to - in one or more
stages are removed. These form a Smale horseshoe as described
below, fig 6(b). This is in fact the Julia set of the mapping,
which in this case is not a connected set, because of the destruction
of the finite attractor's basin.

(6) External noise : In the presence of external noise, the
higher periods become lost, leaving noisy low periods and chaos.
Noise thus can mask high period attractors and create the impression
of chaos, see fig 7.

(ii) The Transition from Quasiperiodicity
to Chaos. A third route to chaos arises from the development
of multiple ( in particular three ) frequencies through repeated
Hopf bifurcations as illustrated in fig 5(a).

The Hopf bifurcation creates an oscillation by the formation
of a cyclic closed orbit as in fig 1(b). Here the vector field
F expressed in terms of polar coordinates (r,q) has a constant
rate of rotation, and a quadratic radial component dependent on
v. For v < 0 the radial component is negative for
all r and hence the origin is a sink (attractor). As v crosses
0 a positive radial component develpos. An attracting periodic
closed orbit attractor (oscillation) is created, leaving a source
(repellor) at the origin, as represented in 1(b) for the value
v = 1.

The quasiperiodicity route is common in the development of
turbulent phenomena through oscillations. The first and second
Hopf bifurcations introduce two frequencies which can be realized
as a flow on a 2-torus to which the rest of the flow is attracted,
fig 5(a). This can be conveniently studied using a Poincare map
:[1.26]

which maps a cross-section C of a flow into itself in the neighborhood
of a periodic orbit by following the flow until it intersects
the cross section again, fig 5(c). This results in an iteration
on the cross section C in which points are mapped to their positions
one cycle later. This is sometimes called a phase portrait because
the mapping arises from the effect of a phase shift in the oscillation
on the closed orbit.

The Poincaré map of a two-dimensional flow on the torus
with angular frequencies and results in a rotation
of the circular cross section by an angle . Adding
a small perturbation we get : [1.27]

If is rational, i.e. a, b integer, the orbits
are periodic and meet themselves again, fig 5(b) after b cycles
through C. However if W is irrational, the orbits pass arbitrarily
close but never meet, fig 5(c) and are called quasi-periodic.
Each orbit is then dense and ergodic on the whole torus.

Further bifurcation to form a third frequency will generally
lead to collapse of the corresponding 3-torus to form a strange
attractor. Like the intermittency route, this results in chaos
after only a finite sequence of bifurcations. It differs from
the other two however in the increase in the dimensionality of
the attractor with each bifurcation.

Fig 5 : (a) Repeated Hopf bifurcations result
in tori. Creation of two oscillations results in a flow on the
2-torus. (b) Periodic flow on 2-torus results in closed orbits
which meets themselves exactly. (c) Poincare map of a cross section
maps each point in a cross section C to the corresponding point
one cycle on along the flow. The flow illustrated is irrational
and hence has orbits consisting of lines which do not meet themselves,
but cover the torus ergodically passing arbitrarily close as time
increases. (d) Breakup of the torus under the circle map as K
crosses 1. The increasing energy thus disrupts the periodic relationships
as chaos sets in. (e) f(q) versus q for the circle map. At K =
.7 the function is 1 - 1 and hence invertible, but for K = 1.6
it is not. (f) The devils staircase of mode-locked states. These
order the possible rationals assigning to each the interval of
values for which such mode-locking occurs for K = 1. At this value
the mode-locked states fill the interval, leaving only a Cantor
set of irrational flows. (g) K -
diagram of the circle map showing mode-locked tongues (K<1)
and chaos densely interwoven with periodicity (K>1). The rational
mode-locking exist only on the curves for K > 1.

Study of this breakup can be facilitated by examining the dissipative
circle map derived from a periodically excited rotator (Schuster
1986): [1.28a]

[1.28b]

In fig 5(d) is shown the cross section of the torus determined
by this map and its breakup as K crosses 1.

The variables are .

This map reduces with b0 (high dissipation) to:
[1.29]

This is simply a special case of the circle map of [1.27].
The sin term can be replaced by any periodic function which possesses
the transition shown in fig 5(e) from a 1-1 function which is
invertible, to a non-invertible form.

The development of chaos with increasing K is as follows (
see fig 5(f,g) ) :

(1) Mode locking : As K varies from 0 towards 1, a set of intervals
of relative frequency occur on which the dynamic
is mode-locked into rational frequency relationships, called Arnold
tongues. Between these there is an irrational flow. Both irrational
and rational cases have non-zero measure. Universal scaling properties
similar to [1.25] occur locally for n values approaching
the golden mean and globally for the tongue widths.

(2) Devil's staircase : At K = 1 these tongues fill the interval,
leaving only a Cantor set of values of measure zero and
fractal dimension 0.85 with irrational dynamics. These form the
Devil's staircase of ordered rational values as shown in (f),
in which successive rationals each have an interval over which
resonance occurs.

(3) Chaos & Order : For K > 1 chaotic and non-chaotic
regions are densely interwoven in K-parameter space.
This means that any state neighbors both chaotic and quasi-periodic
states. For each rational mode-locked state there are two curves
in parameter space which retain the cycle length of the mode-locked
case.

(iii) Conservative Systems
and the Mixing Process. The involvement of chaos in turbulent
dissipative systems does not stop conservative systems displaying
chaotic dynamics. In particular, in conservative dynamical systems,
the lack of attractors leads to structurally unstable configurations
in which chaos and quasi-periodic motion can coexist in the same
system depending on the initial conditions. In fig 6(a) a single
value of the parameter gives both periodic orbits (ellipses) and
chaotic orbits (stippled areas).

Conservative systems have similar mode-locking to the dissipative
case, except that here the rational frequencies give rise to fractal
Cantor-tori and chaos. In fig 6(a) below, is shown the Chirikov
map which forms the discrete integral of a conservative rotator
periodically kicked by a sinusoidal potential: , [1.30]

equivalent to the Poincaré map of a continuous system.
Poincare maps of conservative systems generally include homoclinic
(self-seeking) or heteroclinic orbits joining unstable saddles
at hyperbolic fixed points, as in fig 6(c). Hyperbolic and elliptic
fixed points run vertically down 6(a) forming X and O centres
respectively. For K < 0.972, the orbits with momenta between
p- & p+ remain confined and separate the chaotic stippled
regions on either side. For k = 1.13, a different pattern emerges
with the chaotic region becoming joined by a fractal boundary.
This makes possible a phenomenon called deterministic diffusion,
in which the momentum can wander in value with time.

A final important attribute of chaotic systems is the fractal
nature of the mixing process. In the two-dimensional illustration
of the Smale horseshoe (Holmes 1988), fig 6(b), a map is approximately
linear in two regions U & D which are mapped firstly by a
linear squeezing of the square horizontally and then a linear
stretching and folding over, so that U is mapped on to R and D
onto L. Hence a portion of L(=0) is mapped over each of L &
R, and similarly for R(=1). This results in a fractal Cantor subset
which is mapped indefinitely within itself. Because an element
of this subset can be associated with every infinite sequence
0101110. . . etc. , the structure must contain periodic orbits
of every period, such as for example 001001. . ., non-periodic
orbits corresponding to any random sequence, and even a single
dense orbit which can be constructed by writing the binary digits
in sequence 0.1.10.11.100.101. . . Note also that the Liapunov
exponents 1 contracting, and 2 stretching guarantees
that 1 < 1 < 2 causing an unstable
saddle. A homoclinic saddle as in (c) is generally associated
with a horseshoe, which in turn is an indicator of chaos. The
logistic map for r > 4 provides a one-dimensional example of
a horseshoe Cantor set, fig 2(d).

carries out a very similar process to the horseshoe construction.
It can be decomposed into an area-preserving bending, followed
by lateral contraction and a rotation, as illustrated in fig 6(d).

(d) Quasiperiodicity, Stochasticity
& Chaos

It is important to be able to distinguish chaotic systems from
systems which may have both multiple periodicity and a degree
of external noise or stochastic behavior. We have seen that the
presence of external noise suppresses the higher-order periodic
attractors, thus requiring further tests to eliminate hidden periodicities.

The power spectrum is one measure of the difference between
chaos and quasi-periodic motion. The Liapunov exponent can also
be used. In a purely stochastic process in which xn is subsequently
distributed randomly across all possible values of xn+1,
the Liapunov exponent will be . By contrast, in a quasi-periodic
system with many periodic attractors, the Liapunov exponents should
all be less than 1. For chaotic systems some of the Liapunov exponents
should have positive finite values. These can be calculated indirectly
as follows : We firstly use an xn+1, xn plot of a time-series to build up a profile
of the transfer function xn+1 = G(xn). We then use this graph to make an empirical
calculation of G'(x). Finally we can use [1.7] to get l (see fig
13(d)). Existence of chaos can also be established by demonstrating
a period 3 orbit, fig 13(d) and applying the previously result
that period 3 orbits imply chaos. Because the quasiperiodicity
route to chaos requires only three interacting frequencies, non-linear
systems with multiple frequencies have a high probability of entering
a chaotic regime.

The attractor dimension also gives a measure as uncorrelated
random variables should have correlation dimension . In practice
however, because the variables are not completely uncorrelated,
dimensions under about 7 provide evidence for deterministic chaos
as opposed to purely stochastic behavior fig 16(a). Great care
has to be taken however to distinguish chaos from quasi-periodic
signals with perturbing noise.

Singular value decomposition of the matrix of vectors xn =
xn, ... , xk+n can reveal periodicities or chaos in seemingly
random time series. The matrix

can be diagonalized U, W orthogonal : [1.32]

Where the diagonal entries in S satisfy s1 > s2, > ...
> sk > 0 and U is a rotation in k-space ( Albano et. al.
1986b) Hence applying this rotation to the vectors xn
to get should not change the correlation dimension.

Since the rotated vectors [1.33]

in the event that the singular values si have an abrupt order
of magnitude decrease after sj then the rotated vectors can be
reduced to their first j components. This method can sometimes
unveil low-dimensional or quasi-periodic dynamics with perturbing
noise as opposed to high-dimensional noise, as illustrated in
fig 7 in which rotation using singular decomposition reduced the
dimension from unsaturated (d2>7) to about 2.6 (Albano et.
al. 1986b).

Singular value decompositions can under suitable conditions
be used on their own to estimate dimensions, but several systems
such as the logistic and Henon maps do not give easily-separated
singular values (Mees et. al. 1987). An alternative strategy to
improve time series estimates of correlation dimension is to use
singular-value decomposition to rotate the vectors and choose
a cut-off in the singular values at for example 10^-4 (Albano
et. al. 1988) and subsequently vary the window length in the log-
log
plot to maximize the slope plateau.

(e) Chaos at the Quantum Level
and Reduction of the Wave Packet. Quantum systems differ
fundamentally from the classical case. While the evolution of
the system proceeds according to a deterministic Hamiltonian equation
: [1.34]

creation and destruction of quanta, particularly in the measurement
process, result in causality violations in which the probability
interpretation [1.35]

constitutes the limits on our knowledge of the system. This
results in a stochastic-causal model, in which measurement collapses
the wave function from a superposition of possible states into
one of these states. While quantum-mechanics predicts each event
only as a probability, the universe appears to have a means to
resolve each reduction of the wave-packet uniquely, which I will
call the principle of choice. This is the subject of Schrödinger's
famous cat paradox, in which quantum mechanics predicts a cat
killed as a result of a quantum fluctuation is both alive and
dead with certain probabilities, while we find it is only one
: alive, or dead!

Repeated attempts to model a variety of quantum analogues of
classical chaotic systems have revealed significant differences
which prevent the full display of chaotic dynamics. For example,
the quantum kicked rotator, the analogue of the Chirikov map displays
two types of solution, one rationally periodic with a parabolic
gain in energy, and the other irrational with only non-diffusive
almost-periodic motion. Quantum tunneling (Giesel et. al. 1986)
and level repulsion (Schuster 1986) both tend to inhibit the chaotic
dynamics of such systems .

The case of the Hydrogen atom in a microwave field (Casati
et. al. 1986, Pool 1989) gives the closest approximation to chaos,
including quantum diffusion. However numerical simulation of the
quantum system remains entirely time-reversible and will regain
the initial conditions, for example by phase reversal of the Fourier
expansion, unlike the non-reversibility of the classical solution.
Laser stimulation of molecules such as acetylene (Pique et. al.
1987) also displays borderline chaos in the fine spectra, supporting
the notion that more complex molecules may display quantum chaotic
phenomena under stimulation.

However it is the stochastic wave-reduction
aspect of quantum mechanics which appears to underpin the uncertainty
found in classical chaos. The statistical mechanics of molecular
systems ultimately derives its randomness from Heisenberg uncertainty:

[1.36]

in the form of wave-packet reduction. The position of a molecule
is thus uncertain as a result of the spreading of its wave function.
This uncertainty is unstably reflected in subsequent kinetic encounters
causing e-small perturbations of a classically chaotic system.
One of the important roles of classical chaos may thus be the
amplification of quantum uncertainty into macroscopic indeterminacy.
Ultimately sensitive-dependence in classical systems will result
in quantum inflation, the amplification of quantum fluctuation
into global perturbations of the dynamic.